Abstract
In this paper we study the problem of finding a small safe set S in a graph G, i.e. a non-empty set of vertices such that no connected component of G[S] is adjacent to a larger component in \(G - S\). We enhance our understanding of the problem from the viewpoint of parameterized complexity by showing that (1) the problem is W[2]-hard when parameterized by the pathwidth \(\mathsf {pw}\) and cannot be solved in time \(n^{o(\mathsf {pw})}\) unless the ETH is false, (2) it admits no polynomial kernel parameterized by the vertex cover number \(\mathsf {vc}\) unless \(\mathrm {PH} = \varSigma ^{\mathrm {p}}_{3}\), but (3) it is fixed-parameter tractable (FPT) when parameterized by the neighborhood diversity \(\mathsf {nd}\), and (4) it can be solved in time \(n^{f(\mathsf {cw})}\) for some double exponential function f where \(\mathsf {cw}\) is the clique-width. We also present (5) a faster FPT algorithm when parameterized by solution size.
Highlights
IntroductionThe second result implies that there is no polynomial kernel parameterized by solution size, as the vertex cover number is an upper bound on the size of the solution
For a vertex set S ⊆ V (G), we denote by G[S] the subgraph of G induced by S, and by G − S the subgraph induced by V \ S
We study the problem of finding a small safe set mainly from the parameterizedcomplexity point of view
Summary
The second result implies that there is no polynomial kernel parameterized by solution size, as the vertex cover number is an upper bound on the size of the solution. Their hardness proof implies that the parameters are hard to approximate within a factor of 1.3606 ([1]) They showed that a minimum connected safe set in a tree can be found in linear time. Agueda et al [1] studied their unweighted versions They presented an XP algorithm for graphs of bounded treewidth and showed that the problems can be solved in polynomial time for interval graphs, while they are NP-complete for split and bipartite planar graphs of maximum degree at most 7. Observing that the treewidth of a graph is bounded by a function of its safe number, they showed that the problems are FPT parameterized by solution size
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